Bluetooth DACs with pretty good audio quality and acceptable latency exist, but something including a proper microphone is a bigger challenge.
Two way Bluetooth is out of the question due to low quality codecs and questionable stability across different systems, solutions with USB receivers or optionally something that would normally be used in a studio/stage environment would be probably the way to go.
But even the best rated headphones (Steelseries, Bose, Jabra) with boom microphones offer microphone audio quality from mediocre to okay. (you can find recordings on rtings.com) They just sound thin and boxy, when compared to a condenser microphone the difference is massive. I can accept some tradeoffs (not asking to bend physics), but those differences make the voice simply unpleasant to listen to.
Any tips from the community? Maybe some hacks and something I could use not exactly meant for that purpose?
The best I can think of for now is a high quality Bluetooth DAC (Fiio, Qudelix) for the output and some sort of a wearable / lavalier microphone transmitter meant for video recording (that can be treated as an USB audio interface), but ideally I'd like to have that combined into a single tx/rx unit.
It's not clear how you'd find a headset that supports this, but this is the terminology to look for.
https://www.bluetooth.org/DocMan/handlers/DownloadDoc.ashx?d...
As you mentioned, the best way to go is with something made for the studio like a lavalier and transmitter.
Quality, while probably not studio-grade, is definitely better than shitty Bluetooth.